围长至少为5的图上自旋系统的谱局部到全局原理
A Spectral Local-to-Global Principle for Spin Systems on Graphs with Girth At Least Five
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中文总结 AI 辅助
该研究针对围长至少为5的图，提出谱局部到全局原理，将真q着色Glauber动力学快速混合结果推广到满足局部谱收缩条件的多自旋系统，证明过程结合了与GPT-5.6 Sol Ultra的交互思路。
中文摘要 AI 辅助
证明了对于任意δ∈(0,1)，当q≥(1+δ)Δ且基础图的围长至少为5、最大度Δ=Ω_δ(1)时，真q着色均匀分布的Glauber动力学是快速混合的。该结果还可推广到满足局部谱收缩条件的一般多自旋系统，包括q≥(1+δ)(1-β)Δ的反铁磁Potts模型。这些结果通过针对围长至少为5的图上一般多自旋系统的新谱局部到全局原理，以及星形图上Glauber动力学的新型傅里叶分析得到。所有证明的核心思路是通过与GPT-5.6 Sol Ultra的多轮交互形成的。
英文摘要
It is proved that, for every $δ\in (0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q \geq (1+δ)Δ$ and the underlying graph has girth at least $5$ and maximum degree $Δ= Ω_δ(1)$. This result also extends to general multi-spin systems satisfying a $\textit{local spectral contraction}$ condition, including the anti-ferromagnetic Potts model with $q\geq (1+δ)(1-β)Δ$. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.
